Local renormalizable gauge theories from nonlocal operators
Introduction
The understanding of the behavior of Yang–Mills theories in the nonperturbative infrared region is a great challenge in quantum field theory. Different approaches are currently employed to address this issue, namely: lattice gauge theories,1 study of the Schwinger–Dyson equations [2], [3], [4], [5], duality mechanisms [6], [7], [8], [9], [10], restriction of the domain of integration in the Feynman path integral in order to take into account the existence of the Gribov copies [11], [12], [13], variational principles [14], [15], condensates [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], exact renormalization group [26], [27]. Several results have been achieved so far, having received confirmation from the various approaches. This is the case, for example, of the infrared suppression of the two point gluon correlation function and of the infrared enhancement of the ghost propagator in the Landau gauge [11], [12], [13], [2], [3], [4], [5], [26], [27], [28], [29], [30], [31], [32], [33], [34]. Nevertheless, a satisfactory description of the gluon and quark confinement is not yet at our disposal. One still has the feeling that much work is needed.
The aim of this paper is to call attention to the fact that there exist nonlocal operators which can be consistently added to the Yang–Mills action. This means that, for those specific operators, a renormalizable computational framework can be worked out. As is well known, adding a nonlocal term to the Yang–Mills action is a delicate operation. In most cases the requirement of renormalizability cannot be accomplished. However, in a few cases, the nonlocal term can be cast in local form through the introduction of additional localizing fields. Furthermore, the resulting local theory might exhibit a rich content of symmetries, enabling us to establish its multiplicative renormalizability to all orders. It is worth underlining that, being nonlocal, these operators can induce deep modifications on the large distance behavior of the theory. As such, they might be useful in order to investigate nonperturbative features, being of particular interest for confining theories.
As an explicit example of such nonlocal terms, we shall present a detailed analysis of the nonlocal operatorwhere D2 stands for the covariant Laplacian
Through this example we shall be able to provide a general overview of what can be called a consistent framework [38], [39] for a nonlocal operator which can be added to the Yang–Mills action, namely:
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achievement of a localization procedure,
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investigation of the symmetry content of the resulting local action,
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proof of the multiplicative renormalizability of the theory.
In order to have an idea of the relevance of such nonlocal terms for the infrared behavior of Yang–Mills theories, let us spend a few words on two examples which have been analyzed recently, and which fulfill the requirements of localizability and renormalizability. The first example is provided by the Zwanziger horizon term which implements the restriction of the domain of integration in the Feynman path integral to the Gribov region Ω in the Landau gauge2[12], [13], namelywhere the parameter γ, known as the Gribov parameter, has the dimension of a mass. The second example is given by the gauge invariant nonlocal operatorwhich, when added to the Yang–Mills action, yields an effective gauge invariant mass m for the gluons [37], [38], [39], a topic which is receiving increasing attention in recent years. As shown in [12], [13], [40], [41], [38], [39], both operators (3), (4) are localizable, the resulting local theories enjoy the property of being renormalizable. In particular, in [42], [43], [39] one finds the two loop calculation of the anomalous dimensions corresponding to expressions (3), (4).
Concerning now the operator , Eq. (1), a few potential interesting features might be pointed out in order to motivate better its investigation. We observe that its introduction in the Yang–Mills action leads to a deep modification of the gluon propagator in the infrared. More precisely, as we shall see in the next section, the addition of the term (1) will give rise to a tree level gluon propagator which is of the Gribov type [11], [12], [13], i.e., it is suppressed in the infrared, exhibiting positivity violation, a feature usually interpreted as a signal of confinement. This should be not surprising. Notice in fact that, in the quadratic approximation, both operators (1), (3) reduce to the same expression, thus yielding the same propagator. Also, we mention that expression (1) can be easily adapted to the lattice formulation, thus it could also be investigated through numerical simulations.
The present work is organized as follows. In Section 2, we describe the localization procedure for the operator (1). Section 3 is devoted to the study of the symmetry content of the resulting local action. In Section 4, we derive the set of Ward identities. In Section 5, we present the algebraic characterization of the most general local invariant counterterm, and we establish the renormalizability of the model. Section 6 collects the conclusion.
Section snippets
The localization procedure
Let us start by considering the gauge fixed Yang–Mills action with the addition of the nonlocal operator , Eq. (1), namelywhere SYM is the Yang–Mills action in four dimensional Euclidean space–time,withThe term Sgf stands for the gauge-fixing term, here taken in the Landau gauge, i.e.,where the auxiliary field ba is the Lagrange multiplier enforcing the Landau gauge condition, , and are
Symmetry content
To analyze the symmetry content of our model we shall start first by considering the case in which the parameter σ is set to zero, i.e., σ = 0, yieldingThe action (14) is completely equivalent to the Yang–Mills action, since the introduction of the auxiliary fields and amounts simply to inserting a unity factor, i.e.,Furthermore, the action (14) enjoys the following symmetries
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The BRST symmetry:
Ward identities
The action (46) enjoys a large set of Ward identities. In fact, it turns out that Σ fulfills:
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The Slavnov–Taylor identity
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The Landau gauge-fixing condition
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The anti-ghost equation
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The ghost equationwhereandNotice that
Algebraic characterization of the invariant counterterm and renormalizability
Having established all Ward identities fulfilled by the complete action, Eq. (46), we can now turn our attention to the characterization of the most general invariant counterterm ΣCT. Following the algebraic renormalization procedure [44], ΣCT has to be an integrated local polynomial in the fields and sources with dimension bounded by four, with vanishing ghost and numbers as well as -charge, and obeying the following constraints
Conclusion
Nonlocal operators are known to play an important role in Yang–Mills theories. For example, in the absence of quarks, the vacuum expectation value of the Wilson loop proves to be an order parameter for the confining and deconfining phases of Yang–Mills theories. Also, for a large class of loops, the Wilson operator is renormalizable.
In this work we call attention to the existence of a slightly different class of nonlocal operators which can be added to the Yang–Mills action, while leading to a
Acknowledgments
The Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil), the SR2-UERJ and the Coordenação de Aperfeiç oamento de Pessoal de Nível Superior (CAPES) are gratefully acknowledged for financial support.
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